J. M. Casas

CLASSIFICATION OF SOLVABLE LEIBNIZ ALGEBRAS WITH NULL-FILIFORM NILRADICAL

In this article we classify solvable Leibniz algebras whose nilradical is a null-filiform algebra. We extend the obtained classification to the case when the solvable Leibniz algebra is decomposed as a direct sum of its nilradical, which is a direct sum of null-filiform ideals and a one-dimensional complementary subspace. Moreover, in this case we establish that these ideals are ideals of the algebra as well.

Universal Strict General Actors and Actors in Categories of Interest

For any category of interest ℂ we define a general category of groups with operations

On universal central extensions of Hom-Leibniz algebras

\mathbb{C_G}, \mathbb{C}\hookrightarrow\mathbb{C_G}

Noncommutative Leibniz–Poisson Algebras

, and a universal strict general actor USGA(A) of an object A in ℂ, which is an object of

On Solvability and Nilpotency of Leibniz n-Algebras

\mathbb{C_G}

The actor of a crossed module in lie algebras

. The notion of actor is equivalent to the one of split extension classifier defined for an object in more general settings of semi-abelian categories. It is proved that there exists an actor of A in ℂ if and only if the semidirect product

Universal central extensions in semi-abelian categories

{\text{USGA}}(A)\ltimes A

ON NILPOTENT LEIBNIZ n-ALGEBRAS

is an object of ℂ and if it is the case, then USGA(A) is an actor of A. We give a construction of a universal strict general actor for any A ∈ ℂ, which helps to detect more properties of this object. The cases of groups, Lie, Leibniz, associative, commutative associative, alternative algebras, crossed and precrossed modules are considered. The examples of algebras are given, for which always exist actors.

Crossed modules for Leibniz n-algebras

We develop a theory of universal central extensions of Hom-Lie algebras. Classical results of universal central extensions of Lie algebras cannot be completely extended to Hom-Lie algebras setting, because of the composition of two central extensions is not central. This fact leads to introduce the notion of universal

Homology with Trivial Coefficients of Leibniz n-Algebras

\alpha